sign in sign up
<<
Home , Farad, Gaussian units, Planck units, Statvolt

arXiv:1512.05394v2 [physics.class-ph] 30 Dec 2015 ∗ † ihi .Iao.e-mail: Ivanov. G. Mikhail h ae setne eso fteppr[] ulse nR in published [1], paper the of version extended is paper The rvdsacac oiiit rda nrdcino h Ph the Keywords of introduction SI. gradual revisio the initiate of future to possible chance the a ¾On provides are Resolution field (2011) electromagnetic the CGMP for XXIV units equations the the System and Technology sys units, and SI Gaussian Physics the the In of suggested. modification is A field. electromagnetic for 2 ]adppr[4]. paper and 3] [2, dispute century half a and One Introduction. 1. relativity. utpe fCS nt teCS soeo l n o obsolet now and applications. old practical of for Electri one small is of CGSM too (the Congress were originat units International SI CGSM of First of multiples units The Electrodynamical at (SI). established Units of System hi of of practice Overview in implementation its and contemporary electrodynamics of of improvement The (1907–1909). final Minkowski its H. acquired Electrodynamics (1861–1873). J. o omleadaayia calculations. analytical and formulae public for scientific T for physicists. standard a among is popular CGS very The is electrodynamics. (CGS) CGS-system rical) rfrne r omnntol o h lsiso science, of classics Technology. the and “Basi for Physics us Tamm only are of I.Ye. SI) Institute not by the common of textbook are units classical electrical preferences in of predecessors used (the is units CGS the E.g. ytmo nt S) 6 dis(UisotieteS” the SI”, the outside (“Units M.I.) admits by 8-t [6] font last (SI)” the Units in of Measures System and Weights of Bureau International h otmoaypatc st ao h s fteS nt f units SI the of use the favor to is practice contemporary The eilto n tnad fmn onre eomn ous to recommend countries many of standards and Legislation bac dates electrodynamics for units of choice of problem The n ol wt oeeagrtn)dcaeta h Ii sys is SI the that declare exaggerating) some (with could One stil is electrodynamics in SI the of using the Nevertheless, oa h tnadfreetia icisi h I u for but SI, the is circuits electrical for standard the Today h rbe fcoc fuisfreetoyaisi tl a still is electrodynamics for units of choice of problem The HSC N EHOOYSSE FUISFOR UNITS OF SYSTEM TECHNOLOGY AND PHYSICS I nentoa ytmo nt,eetoyais Phys electrodynamics, Units, of System International SI, : some ytm fuis hc r sdi lcrdnmc ol be could electrodynamics in used are which units, of systems [email protected] ocwIsiueo hsc n Technology and Physics of Institute Moscow ELECTRODYNAMICS ogpun,Russia Dolgoprudny, .G Ivanov G. M. Abstract 1 ussian. dfrteatraigcret( 0o 5) h similar The [5]). of 80 (§ current alternating the for ed hreuainsrthdntls hnahl fcentury. of half a than less not stretched education gher omol fe emtiaino pca eaiiyby relativity special of geometrization after only form lottesm oma nteGusa ytm The system. Gaussian the in as form same the almost sc n ehooySse sanwmodification new a as System Technology and ysics -dprgah oewr r ihihe yitalic by highlighted are word some paragraph, 2-nd in Prs 81.Uiso rcia ytmare system practical of Units 1881). (Paris, cians tosadtxboso hoeia physics. theoretical on textbooks and ations e fuis(h hsc n ehooySystem) Technology and Physics (the units of tem dto fisoca rcue“h International “The brochure official its of edition h fmaueetfreetia iciscicd with coincide circuits electrical for of sfo h rcia ytmo nt,wihwas which units, of system practical the from es u o oa tdnsadpoesr nMoscow in professors and students today for but atro betos h asin(symmet- Gaussian The objections. of matter a l 4dmninlrltvsi oain)formulation covariant) relativistic (4-dimentional † reeti icisadteGusa G system CGS Gaussian the and circuits electric or eCSi oecnitn ihsmere of symmetries with consistent more is CGS he lcrmgei edtesadr steCGS. the is standard the field electromagnetic fteItrainlSse fUis h SI¿ the Units, of System International the of n ta,mroe,i eoe oeata.The actual. more becomes it moreover, ctual, otetm fM aaa 12–81 and (1822–1831) M. of the to k e o esrmnsadteCSi system is CGS the and for tem hoyo lcrct”[] u h practical the but [5], ” of theory c nsineadeuainTeInternational The education and science in e eso fteCSsse fuis,which units), of system CGS the of version e c n ehooySse fuis special units, of System Technology and ics ∗ on,freape nbooks in example, for found, Individual scientists should also have the freedom to sometimes use non-SI units for which they see a particular scientific advantage in their work. An example of this is the use of CGS- in electromagnetic theory applied to quantum electrodynamics and relativity. Reading this text one has to remember, that coherent modern representation of the electrodynamics without is impossible. It is clear, that the main message of this disclaimer applies primarily to usage of the CGS system. The previous 7-th edition of the brochure “The International System of Units (SI)” [7] does not contain this disclaimer. The softer attitude towards the CGS system is probably associated with the planned change of definition of , , and in the SI by fixing exact numerical values of the Plank constant, the , the Avogadro constant and the [8]. This redefinition requires new exact measurements and calculations, which use methods of quantum electrodynamics. Just in quantum electrody- namics the CGS looks preferable in comparison with the SI for many physicists.

2. Criticism of the SI

Usage of the SI in theoretical research is complicated. The SI-units of electrodynamic quantities do not consistent with symmetries of the theory, which related with special relativity. This inconsistency is related with different dimensions of the electric and magnetic fields E and H and the inductions D and B. It makes difficult usage of the SI in the teaching of electrodynamics, especially in the cases where a student has to have a solid understanding of the theory structure. Because of this reason courses in leading Russian universities traditionally use the CGS. The problem was considered in paper by D.V. Sivukhin “The international system of physical units” [9] (see also [10], where the comparison of the SI and the CGS is presented), published in 1979 in the journal “Soviet Physics Uspekhi” by the decision of the Division of General Physics and of the USSR Academy of Sciences. That practically means the unified position of community of Russian physicists. We present a vivid quote from this article: In this respect, the SI system is no more logical than, say, a system in which the , width, and height of an object are expressed not only in terms of different units but have different dimensions as well. The inconsistence of the SI with the symmetries of electrodynamics due to historical reasons, because the foundations of the system were set before the creation of special relativity. Moreover, the units of , ampere, , , etc. (ascending to the practical system) are intensively used in technics, are involved in the SI and are not involved in the CGS1. When in 1948 these units were introduced in the SI, role of the special relativity for electrodynamics was still not sufficiently understood by many physicists-experimenters and engineers. You could ever dream of how to change these units in the early XX century, but now this units and related standards are widely used not only in measuring devices, but in the whole technique, including home electronics. This makes it virtually impossible any revision of the SI, which excludes from the system ampere, as the basic unit of. Can we combine the wishes of the engineers and theorists? Yes, we can!

3. What system of units do we need? 3.1. The wishes of a theorist Electric field strength E and induction D have to be of the same dimension, and in the E = D. • Electric field induction B and strength H have to be of the same dimension, and in the vacuum B = H. • Fields E and B have to be of the same dimension, and in the vacuum for flat traveling wave E = B . • v | | | | Magnetic field of a moving charge is a relativistic effect, so the formula have to contain c . To remove speed • of by redefinition of units is unnatural.

1In the beginning of XX century in Russia were fabricated, labeled in the CGS system. The CGS unit of is the centimeter [11].

2 v The Lorentz is a relativistic effect, so the formula have to contain c . To remove by • redefinition of units is unnatural. 1 Introducing of constant 4π into the law (it was suggested by O. Heaviside) is natural, because • it removes the factor 4π (the surface of two-dimensional unit sphere) in Maxwell equations and in formulae for and action of electromagnetic field. It is consistent with the practice of theorists which consider spaces with dimensions other then 3. For the standard electrodynamics this rationalisation makes no bad nor good.

3.2. The wishes of an engineer and an experimenter The SI units for electrical circuits (ampere, volt, ohm, farad, ) are used everywhere in devices and • standards, they could not be changed. The appearance of the speed of light in equations for electrical circuits is undesirable. • Fields D and H could not be measured directly, so their units are not used in any devices. This units are • not very important.

3.3. How to reconcile theorists with engineers We propose to modify the SI without changing the base units (kilogram, , , ampere, mole, kelvin, ), but modifying the form of the equations of electrodynamics (to make them similar to equations in the CGS) by changing the constant factors. Due to this change of factors some SI derived units have to be changed. We preserve from old SI (SIold) ampere as unit for current, and all derived units, which do not involve fields D, B and H. So the units of charge coulomb = C, of volt = V, of electrical resistance ohm =Ω, of capacitance farad = F, of henri = H remain the same. The Coulomb law and the strength of electrical field reman the same form as in SIold

1 q1q2 q1q2 1 q q F = 2 = ke 2 , E = 2 = ke 2 . 4πε0 r r 4πε0 r r

1 1 2 Here ke = 4πε0 is , q , q , q are electrical charges. We redefine the fields D, H and B according to above “The wishes of a theorist”, all these fields have the V same unit m , just like field E. All formulae and units for electrical circuits (without fields) preserve the same form as in the SIold. All formulae (but not units!) for electromagnetic fields look similar to formulae of the CGS. This future version of the SI we call Physics and Technology System of units (PT). The name of the PT system based on the idea to combine the advantages of systems used in physics and engineering. During the process of preparing the paper it was discovered that this system of units has already been proposed previously (from 2001) by G.M. Trunov [3, 12, 13, 14]. He calls it theoretical system of electromagnetic units (SI(T)). Unfortunately this initiative has not yet received wide distribution. We think, to change the name of system of units in accordance with the terminology of G.M. Trunov is not feasible, because for the expansion of the new system its benefits in the convergence of physics and engineering are more important than theoretical advantages (the theorists are quite happy with the CGS).

4. The physics and technology system of units (PT) 4.1. Conversion from the CGS to the PT

The base units of the PT system coincide with the base units of SIold. So, using the standard (SIold) equations for current I, electrical potential U, power dE , energy and for capacitance C and inductance L dt EC EL dq d U q2 LI2 I = , UI = E , R = , = , = , dt dt I EC 2C EL 2

3 one derives the units of q, U, C, L in terms of kilogram, metre, second and ampere. All these units coincide with 2 units of SIold. We derive the electromagnetic field equations in the PT system by modification of the CGS equations. In Gaussian CGS the particular choice of gramme and second as base units is not significant. We are interested in just the form of equations (the choice of constants). So, let us introduce an auxiliary Gaussian system of units MKS (metre, kilogram, second Gaussian, MKSG). All equations of MKSG has the same form as the equations of Gaussian CGS. From the point of view of MKSG the Coulomb constant (hereinafter, the braces denote numerical value of dimensional quantity) 1 2 −7 ke = , ke = c 10 , 4πε0 { } { } × is used just to convert charge units from SI to MKSG

1 q1siq2si q1siq2si q1mksg q2mksg F = 2 = ke 2 = 2 , qmksg = ke qsi. 4πε0 r r r p If the force law for a charge q in electric field E does not contain any coefficients, then

Esi F = qmksgEmksg = qsiEsi, Emksg = . √ke

For electrical charges, currents and field E there is no difference between the PT system and the SIold. In the MKSG, like in the CGS all fields E, D, B, H have the same units. To preserve this property in PT system we use the same conversion factor to convert all four fields from the MKSG to the PT. Also we use the equal conversion factors to convert all field sources (charges, currents, electric and magnetic multipoles). I.e. we make the following substitution into the electrodynamics equations of the MKSG (which look like exactly as equations of the CGS) (q, j,... ) = k (q, j,... ) , mksg e × pt p1 (E, D, B, H)mksg = (E, D, B, H)pt. √ke × So we have only two different conversion factor from the PT system to the usual for theorists Gaussian system (not CGS, but MKSG): one conversion factor for all fields (E, D, B, H), the other for all sources (charges, currents, multipoles). The product of the conversion factors is 13. You don’t need a whole table of conversion factors, you use to convert between the SIold and the CGS!

4.2. Equations in the PT system The electrodynamics equations in the PT system have the following form 1 ∂B div B = 0, rot E = , − c ∂t 1 ∂D 4π div D = 4πk ρ, rot H = + k j, e c ∂t c e c (E, D) + (B, H) S = [E H], W = , 4πke × 8πke EαDβ + BαHβ σαβ = δαβW, 4πke − 1 Φ F = q E + [v B] ,L = ,  c ×  cI D = E +4πk P, H = B 4πk M. e − e 2The only variable related with field here is U. All variables defined in terms of circuit parameters. According to elecromechanical analogy charge q is generalised coordinate, current I is generalised velocity, potential difference U is generalised force, resistance R is 1 old viscous friction constant, inverse capacitance C is spring constant, inductance L is constant. Similarly to SI speed of light does not involved in electrical circuit equations. 3The substitution have to be done only in the fields equations. The form of the equations of circuits and the definitions of multipoles is the same as in Gaussian system.

4 Here ρ and j are of charge and current, W is of field energy, S is the Umov-Pointing vector, σαβ is the Maxwell stress tensor, indices α, β = 1, 2, 3 numerate space coordinates, Φ is the magnetic flux, P and M are electric and magnetic dipoles per unit . The substitution ke 1 reproduce the equations in the Gaussian form. The only exception is inductance, → Φ which sometimes defined in the CGS as Lcgs = I . Nevertheless this is not shortcoming, but the advantage, because it removes speed of light from the electrical circuit equations. The conversion from CGS formulae to PT ones (the insertion of the Coulomb constant) is easily performed using dimensionalities. This conversion is not much more difficult, than the insertion of the Boltzmann constant, which was taken to be 1. The fields in the SIold and in the PT system are converted by the following relations

Dsiold Ept = Esiold , Dpt = =4πkeDsiold , ε0

Hsiold 4π Bpt = cBsiold , Hpt = = cµ0Hsiold = keHsiold , cε0 c

1 1 4πke ε0 = , µ0 = 2 = 2 . 4πke c ε0 c To comply with the old system SIold it is useful to introduce an “effective” magnetic field, which coincides with the magnetic field of SIold B B = pt = B eff c siold The modern (SIold) devices are calibrated in the units of effective magnetic field, however, the use of it in formulae is undesirable.4.

5. Conclusion

In 2011 The General Conference on Weights and Measures (CGPM), at its 24th meeting, adopted the Resolution “On the possible future revision of the International System of Units, the SI” [8]. The resolution recommends to change the definition of kilogram and ampere by simultaneous fixing the exact numerical values of the and of the elementary charge. It means that the base SI units will be defined by quantum electrodynamics effects. The 8-th edition of “SI brochure” [6] has already recognized the advantages of “the use of CGS-Gaussian units in electromagnetic theory V applied to quantum electrodynamics and relativity”, so the simultaneous introducing of the same unit m for all four fields E, D, B, H (i.e. the introducing of the PT system as new version of the SI) looks natural and feasible. This reform of the SI would be invisible for the general public, because ampere, volt, ohm, farad and henry remain the same. For the majority of physicists and engineers this reform would not be a big problem, because electrical circuits units remains the same as in the SI, and new field equations are close to the CGS equations. The century dispute on the choice of units for electrodynamics could be resolved in next few years, if the community of physicists to show an active position.

Acknowledgements

The author thanks Yu.R. Alanakyan, A.L. Barabanov, V.S. Bulygin, V.V. Ezhela, A.D. Gladun, A.N Kirichenko, S.M. Kozel, A.S. Leskov, V.A. Ovchinkin, A.A. Pukhov, Ya.A. Romanyuk, A.A. Rukhadze, V.P. Slobodyanin, V.P. Tarakanov, G.M. Trunov, S.V. Vinogradov, V.G. Zhotikov and other faculty of the Department of Theoretical Physics and the Department of General Physics of Moscow Institute of Physics and Technology, and participants of the seminar of the Theoretical Department of Prokhorov General Physics Institute of Russian Academy of Sciences for the useful discussions and valuable criticism.

4It is easy to remember, if one remembers the fact that the written in terms of effective magnetic field does not involve the speed of light.

5 A. The SI in electrodynamics

Let us present for comparison the same key equations of electrodynamics in the SI units: ∂B div B = 0, rot E = , − ∂t ∂D div D = ρ, rot H = + j, ∂t 1 1 S = [E H], W = (E, D)+ (B, H), × 2 2 σ = (E D + B H ) δ W, αβ α β α β − αβ Φ F = q (E + [v B]) ,L = , × I B D = ε0E + P, H = M. µ0 − One could think, that the equations in the SI look simpler. To dispel this illusion, and to explain the reason for the preference of theorists, we write the SI equation in vacuum (fields D and H are excluded)

∂B div B = 0, rot E = , − ∂t 1 ∂E 1 ∂E 1 div E = ρ, rot B = ε0µ0 + µ0j = 2 + 2 j, ε0 ∂t c ∂t ε0c 1 2 ε0 2 1 2 ε0 2 2 2 S = [E B]= ε0c [E B], W = E + B = (E + c B ), µ0 × × 2 2µ0 2 1 2 σαβ = ε0EαEβ + BαBβ δαβW = ε0 EαEβ + c BαBβ δαβW.  µ0  − −  B. The equations of PT system with Heaviside constant

Some physicists prefer the Heaviside-Lorentz units. Let us present the PT system electrodynamics equations with Heaviside constant 1 2 −7 κe = =4πke, κe =4π c 10 . ε0 { } { } × The same key equations of electrodynamics in the PT system with Heaviside constant have the following form 1 ∂B div B = 0, rot E = , − c ∂t 1 ∂D κ div D = κ ρ, rot H = + e j, e c ∂t c c (E, D) + (B, H) S = [E H], W = , κe × 2κe 1 σαβ = (EαDβ + BαHβ) δαβW, κe − 1 Φ F = q E + [v B] ,L = ,  c ×  cI D = E + κ P, H = B κ M. e − e

The substitution κe 1 reproduce the equations in the Heaviside-Lorentz form (in the MKS-Heavisidean system, MKSH). The inverse→ conversion is

(q, j,... ) = √κ (q, j,... ) , mksh e × pt 1 (E, D, B, H)mksh = (E, D, B, H)pt. √κe ×

6 C. The meaning of the Coulomb constant

The tradition of the Gaussian system of units assumes that the Coulomb constant has no physical meaning, since it represents just the square of the ratio of two units of charge:

2 qmksg ke = , { }  1 C  here 1 2 q = √J m = kg / m3/2 s−1 = 104.5 CGS unit of charge mksg · · · is the MKSG unit of charge. In the modern SI, in which ampere (and coulomb as a derived unit) is defined via the force acting between two parallel conductors with a current, the opinion of ke physical meaningless is absolutely true. In 2011 The General Conference on Weights and Measures (CGPM), at its 24th meeting, adopted the Resolution “On the possible future revision of the International System of Units, the SI” [8]. The resolution recommends to redefine kilogram by fixing the exact numerical value of the Planck constant (similarly metre was redefined by fixing the exact numerical value of the speed of light)

h =2π~ =6.626 06X 10−34 J s, × · and to redefine ampere by fixing the exact numerical value of the elementary charge5

−19 e =1.602 17X 10 C. × In both cases X denotes some digits, which have to be fixed in future. In year 1983 the metre was redefined by fixing the exact numerical value of the speed of light. Now by definition speed of light has exact numerical value: m c = 299792458 . s So, in the definition of the fine-structure constant6

e2 k e2 α = = e 4πε0~c ~c all quantities will be defined by exact numerical values by definition, except of one, the Coulomb constant. ~c k = α = C α. e e2 · α ·

We see that the numerical value of ke in the future reformed SI is the fine structure constant, multiplied by some fixed numerical factor. The factor is chosen so that the redefined units of mass and current reproduced taken thus far with the accuracy available with modern (at the time of the reform) measuring devices. The further improvement of the accuracy of the measurements will inevitably makes the difference of units of the modern SI and the reformed SI available for measurement. With this moment the Coulomb constant becomes really fundamental constant. It is the fine structure constant multiplied by historically fixed factor Cα. Thus, the Coulomb constant acquires a physical meaning due to the fact that we have a natural charge unit (the elementary charge), which is not a multiple of Planck charge. The dispute between the Gaussian CGS system (and the old SI) and the future SI on definition of charge unit is related to the following simple choice. Do we prefer to define charge unit using the Plank charge (the CGS system and the old SI), or using the elementary charge (the new SI). Since the Planck charge in is not implemented, the choice seems to be obvious.

5It would be natural to require that 1 C were an integer number of e. 6The fine-structure constant itself is the square of the ratio of two of charge, the elementary charge and the Plank 2 2 = e = e ≈ 1 charge α ~c q 137,035 999 .  cgs  Pl 

7 D. The practical units-1990 and the redefinition of the ampere and the kilogram

This appendix is based mainly on materials of the paper by S.G. Karshenboym “On the redefinition of the kilogram and ampere in terms of fundamental physical constants” [15]. The possible modernizations of the SI are discussed in paper [16]. Simultaneous redefinition of kilogram and ampere by fixing exact numerical values of the Plank constant and the elementary charge is related with the following fact. The modern official definition of ampere using the force of interaction between two parallel currents. It corresponds to fixing of exact numerical value of

ε0 −7 H µ0 = =4π 10 . c2 × m This official definition is less precise than definition of ampere using quantum effects. Quantum effects are used in definitions of “practical” units of year 1990, the volt-1990 and the ohm-1990. The volt-1990 is defined using the Josephson effect. During the flow of superconducting current through Josephson junction the extra energy 2eU received by a Cooper pair from external potential difference is emitted 2e in the form of photons. The photon is ν = h U. The definition of the volt-1990 fixes exact numerical value of the Josephson constant 2e GHz KJ = = 483597.9 . h V90 The ohm-1990 is defined using the integer quantum Hall effect, which is observed in 2-dimensional degenerate electron gas at low in magnetic field perpendicular to 2d gas plane. Across the direction of current, RK a proportional to the current appears Uy = ρxyIx. The Hall resistance is ρxy = n , where n is an integer number. The definition of the ohm-1990 fixes the exact numerical value of the von Klitzing constant h R = = 25812.807 Ω90. K e2 So, the simultaneous fixing of the Josephson constant and the von Klitzing constant defines the exact nu- merical values of the Plank constant and of the elementary charge. The units of volt-1990, ohm-1990, metre and second are the base units of the practical system. The practical system is an alternative version of SI. It has higher precision, but it is different from the SI. The “practical kilogram” differ from the standard kilogram by −7 10 , the “practical constant µ0” is not a fixed value. For some modern measurement the main source of inaccuracy is not measurement itself, but the definition of measurement unit. Practical units are defined more precisely than the SI units, so the redefinition of kilogram and ampere according to Resolution of year 2011 looks useful. It does not mean introducing of volt-1990 and ohm- 1990 as the new SI standards, because since the year 1990 measurement technic has improved. The discrepancy of “practical kilogram” with the standard kilogram by 10−7 is no longer acceptable.

E. The notion of the system of the systems of units

Some systems of units (the SI, the system of practical units) are the systems preferably for measurements and the others (the CGS, the Heaviside-Lorentz system and some other systems) are the “theoretical” systems, preferably for formulae and analytical calculations. The units of theoretical systems are sometimes inconvenient for actual measurements. The convenient method to introduce a theoretical system is to set some constants to be 1. E.g. theorists prefer to set the Boltzmann constant to be 1. It is convenient for a theorist, but it is unusable for an experimenter. Nevertheless, after the derivation of final formula, a theorist could easily reintroduce the Boltzmann constant using the dimensionalities of physical quantities. Similarly in the special relativity system of units the speed of light is set to be 1, in the system the too. In the quantum mechanical system the Plank constant ~ is set to be 1, in the atomic physics system the elementary charge and the electron mass too. In all these cases the inverse conversion to experimental units could be done easily by using the dimensionalities of physical quantities.

8 We have the natural system of the systems of theoretical units, which originates from Gaussian CGS7. The other theoretical systems (convenient for different theoretical physics branches) generates from CGS by substituting 1 instead of different constants. And so on until completely dimensionless system of units like the . Unfortunately the form of electrodynamics equations in the CGS and the SIold is different. So the SIold is not connected with the system of theoretical systems. The rules of conversion between the SIold and the CGS are to complicated. The PT system naturally becomes in the root of the system of theoretical systems. To convert from the PT system to the Gaussian system one just set the Coulomb constant to be 1. It connects the systems of theoretical systems with experiment.8

F. RPP-2014 electromagnetic relations

To demonstrate the convenience of the PT system, we reproduce here the section 7 ELECTROMAGNETIC RELATIONS of The Review of [17], using PT units. We expand the table below to add the PT system. Two subsections, which are exactly the same in PT units and in SI units are presented by their headlines and short comments only. The third subsection is rewritten from CSG system to PT system (actually we just insert ke constant in two formulae of the 3rd subsection).

7The other system of the systems of theoretical units originates from the Heaviside-Lorentz system of units. 8The other system of the theoretical systems originates from the Heaviside-Lorentz system is also naturally connected with the PT. To convert from the PT system to the Heavisidian system one just set the Heaviside constant to be 1.

9 7. Electromagnetic relations

Quantity Gaussian CGS SI PT system

Conversion factors:

Charge: 2.997 924 58 × 109 esu =1C=1As =1C=1As

1 1 Potential: (1/299.792 458) (/esu) =1V=1JC− =1V=1JC−

4 4 1 1 8 Magnetic field: 10 = 10 /esu =1T=1NA− m− = 2.997 924 58 × 10 V/m

F E v B F E v B F E v B = q( + c × ) = q( + × ) = q( + c × )

∇ • D = 4πρ ∇ • D = ρ ∇ • D = 4πkeρ

∇ H 1 ∂D 4π J ∇ H ∂D J ∇ H 1 ∂D 4π J × − c ∂t = c × − ∂t = × − c ∂t = c ke

∇ • B = 0 ∇ • B = 0 ∇ • B = 0

∇ E 1 ∂H ∇ E ∂H ∇ E 1 ∂H × − c ∂t = 0 × − ∂t = 0 × − c ∂t = 0

Constitutive relations: D = E + 4πP, D = ε0E + P, D = E + 4πkeP,

H = B − 4πM H = B/µ0 − M H = B − 4πkeM

Linear media: D = ǫE, H = B/µ D = ǫE, H = B/µ D = ǫE, H = B/µ

12 1 1 ǫ0 = 8.854 187 ···× 10− Fm− 1

7 2 1 µ0 = 4π × 10− NA− 1

E ∇ 1 ∂A E ∇ ∂A E ∇ 1 ∂A = − V − c ∂t = − V − ∂t = − V − c ∂t

B = ∇ × A B = ∇ × A B = ∇ × A

′ 3 ′ ′ 3 ′ ′ 3 ′ qi ρ(r ) d x 1 qi 1 ρ(r ) d x qi ρ(r ) d x V = = ′ V = = ′ V = k = k ′ P ri R r r 4πǫ0 P ri 4πǫ0 R r r e P ri e R r r charges | − | ch. | − | ch. | − |

3 ′ 3 ′ 3 ′ A 1 Idℓ 1 J d x A µ0 Idℓ µ0 J d x A ke Idℓ ke J d x = c H r r′ = c R r r′ = 4π H r r′ = 4π R r r′ = c H r r′ = c R r r′ | − | | − | | − | | − | | − | | − |

E′ = E E′ = E E′ = E k k k k k k E E 1 v B E E v B E E 1 v B ′ = γ( + c × ) ′ = γ( + × ) ′ = γ( + c × ) ⊥ ⊥ ⊥ ⊥ ⊥ ⊥

B′ = B B′ = B B′ = B k k k k k k B B 1 v E B B 1 v E B B 1 v E ′ = γ( − c × ) ′ = γ( − c2 × ) ′ = γ( − c × ) ⊥ ⊥ ⊥ ⊥ ⊥ ⊥

1 2 7 2 9 1 µ0 7 2 1 8 1 ke = = c × 10 NA = 8.987 55 ···× 10 m F ; = 10 NA ; c = = 2.997 924 58 × 10 ms 4πǫ0 − − − 4π − − √ǫ0µ0 −

7.1. Impedances (SI units = PT inits) The original text uses SI. The formulae in PT are exactly the same.

10 7.2. Capacitors, inductors, and transmission Lines (SI units = PT inits) The original text uses SI. The formulae in PT are exactly the same.

7.3. Synchrotron radiation (CGS units and PT units)

We use the PT system in this subsection. To convert formulae from PT back to CGS just set ke =1. For a particle of charge e, velocity v = βc, and energy E = γmc2, traveling in a circular orbit of radius R, the classical energy loss per revolution δE is

4π k e2 δE = e β3γ4. (1) 3 r For high-energy electrons or positrons (β 1), this becomes ≈ δE(in MeV) 0.0885[E(in GeV)]4/R(in m). (2) ≈ For γ 1, the energy radiated per revolution into the interval d(~ω) is ≫ 8π dI = αγ F (ω/ω ) d(~ω), (3) 9 c 2 where α = kee /~c is the fine-structure constant and

3γ3c ω = (4) c 2R is the critical frequency. The normalized function F (y) is

∞ 9 F (y)= √3y K5/3(x) dx, (5) 8π Zy where K5/3(x) is a modified Bessel function of the third kind. For electrons or positrons,

~ω (in keV) 2.22[E(in GeV)]3/R(in m). (6) c ≈

For γ 1 and ω ωc, ≫ ≪ dI 3.3α(ωR/c)1/3, (7) d(~ω) ≈ whereas for γ 1 and ω & 3ω , ≫ c

1 2 dI 3π ω / 55 ω αγ e−ω/ωc 1+ + . (8) d(~ω) ≈ r 2  ωc   72 ωc ···  The radiation is confined to angles . 1/γ relative to the instantaneous direction of motion. For γ 1, where Eq. (3) applies, the mean number of photons emitted per revolution is ≫

5π Nγ = αγ, (9) √3 and the mean energy per photon is 8 ~ω = ~ωc. (10) h i 15√3 When ~ω & O(E), quantum corrections are important. h i

11 References

[1] Ivanov M.G. Fiziko-tekhnicheskaya systema edinits dlya electrodinamiki. Inzhen- ernaya fizika, [Physics and technology system of units for electrodynamics. En- gineering Physics] 2015, № 1, pp. 4–12. [http://infiz.tgizd.ru/en/arhiv/13427, https://mipt.ru/students/organization/mezhpr/biblio/razmernost.php] [2] Gladun A.D. Pedagogiechskie razdumya fizika [The pedagogical reflections of the physics]. M.: MFTI [Moscow: MIPT], 2005. 104 p. [3] Trunov G.M. Uravneniya elektromagnitizma i sistemy edinits elektricheskikh i magnitnykh velichin [The equations of and the systems of units of electrical and magnetic quantities]. M.: FORUM [Moscow: Publishing house ¾FORUM¿]. 2011. 104 p. [4] Carron N. Babel of Units. The Evolution of Units Systems in Classical Electromagnetism. arXiv:1506.01951 [5] Tamm I.Ye. Osnovy teorii elektrichestva [Basic theory of electricity]. M.: Nauka. Gl. red. fiz.-mat. lit. [Moscow: Publishing house ¾Science¿ Ch. Ed. Sci. Lit.], 1989. 504 p. [6] The International System of Units (SI), 8th edition (S`evres: BIPM, 2006) [http://www.bipm.org/en/si/si_brochure/general.html] [7] The International System of Units (SI), 7th edition (S`evres: BIPM, 1998) [http://www1.bipm.org/utils/en/pdf/si-brochure.pdf] [8] On the possible future revision of the International System of Units, the SI Resolution 1 of the 24th meeting of CGPM (2011); http://www.bipm.org/en/CGPM/db/24/1/ [9] Sivukhin D.V. O Mezhdunarodnoy sisteme fizicheskikh velichin // UFN 129 P. 335-338 (1979) [Sivukhin D.V. The international system of physical units // Sov. Phys. Usp. 22 P. 834-836 (1979)] [10] Sivukhin D.V. Obshchiy kurs fiziki. T. III. Elektrichestvo. § 85 ¾Mezhdunarodnaya sistema edinits (SI)¿ [General course of physics. T. III. Electricity. § 85 "The International System of Units (SI)"]. M.: FIZ- MATLIT; MFTI [Moscow: Publishing house ¾FIZMATLIT¿; MIPT], 2004. 656 p. [11] Bulygin V.S. chastnoe soobshchenie [Buligin V.S. The private communication] [12] Trunov G.M. Privedenie edinits elektricheskikh i magnitnykh velichin sistemy SI v sootvetstvie s sovremen- nym predstavleniem ob elektromagnitnom pole [Conversion the units of electrical and magnetic quantities of the SI in accordance with the modern idea of the electromagnetic field] // Fizicheskoe obrazovanie v VUZakh [Physics in Higher Education]. 2001. Vol. 7. № 4. P. 12-21. [13] Trunov G.M. Korrektsiya matematicheskoy formy zapisi uravneniy elektromagnetizma i sozdanie na ikh osnove novoy sistemy elektromagnitnykh edinits [The correction of the mathematical form of the equations of electromagnetism and creation on their basis of a new system of electromagnetic units] // Vestnik Moskovskogo gosudarstvennogo tekhnicheskogo universiteta im. N.E. Baumana. Seriya: Yestestvennye nauki [Bulletin of Moscow State Technical University. NE Bauman. Series: Natural Sciences]. 2006. № 2. P. 66-75. [14] Trunov G.M. Uravneniya elektromagnetizma i sistemy elektromagnitnykh edinits  proshloe, nastoy- ashchee, budushchee [The equations of electromagnetism and the electromagnetic system of units - Past, Present, Future] // Zakonodatelnaya i prikladnaya metrologiya [Legislative and applied metrology]. 2006. № 2. S. 46-52. [15] Karshenboym S.G. O pereopredelenii kilogramma i ampera v terminakh fundamentalnykh fizicheskikh konstant // UFN Vol. 176 P. 975-982 (2006) [Karshenboim S.G. On the redefinition of the kilogram and ampere in terms of fundamental physical constants // Phys. Usp. Vol. 49 P. 947-954 (2006)] [16] Fletcher N., Davis R.S., Stock M., Milton M.J.T. Modernizing the SI: implications of recent progress with the fundamental constants. arXiv:1510.08324 [17] K.A. Olive et al. (Particle Data Group) The Review of Particle Physics. Chin. Phys. C, 38, 090001 (2014) [http://pdg.lbl.gov/2014/download/download.html; http://pdg.lbl.gov/2015/reviews/rpp2014-rev- electromag-relations.pdf].

12